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date: 20 September 2017

The electronic structure of epitaxial graphene—A view from angle-resolved photoemission spectroscopy

Abstract and Keywords

This article analyzes the electronic structure of epitaxial graphene using angle-resolved photoemission spectroscopy (ARPES). It first describes how the carbon atoms in graphene are arranged before discussing the growth and characterization of graphene samples. It then considers the electronic structure of epitaxial graphene, along with the gap opening in single-layer epitaxial graphene. It also examines possible mechanisms for the gap opening in graphene, including quantum confinement, mixing of the states between the Brillouin zone corner K points induced by scattering, and hybridization of the valence and conduction bands caused by symmetry breaking in carbon sublattices. Clear deviations from the conical dispersions are observed near the Diracpoint energy, which can be interpreted as a gap opening attributed to graphene–substrate interaction. Graphene–substrate interaction is thus a promising route for engineering the bandgap in graphene.

Keywords: electronic structure, epitaxial graphene, angle-resolved photoemission spectroscopy, gap opening, quantum confinement, Brillouin zone, scattering, conduction bands, symmetry breaking, carbon sublattices

14.1 Introduction

Graphene, the fundamental building block for all graphitic materials, is a two-dimensional sheet of carbon atoms arranged in a honeycomb lattice with sp2 bonding. The sp2 hybridization between one s-orbital and two p-orbitals leads to the formation of σ bonding between carbon atoms, while the π orbitals perpendicular to the plane lead to the formation of the half-filled bands, the π-bands. These half-filled bands are the basis for most of the fascinating properties of graphene, and its full potential is still not reached. Although the theoretical study of graphene started in the 1950s (McClure 1957), the experimental study of graphene had not been realized until the recent discovery and characterization of exfoliated graphene by Novoselov et al. (2004) and epitaxial graphene by Berger et al. (2004). Immediately after that, graphene was found to exhibit various intriguing properties unexpected from conventional materials. For example, the charge carriers in graphene are massless Dirac fermions with mobility higher than that of silicon and the doping can be tuned from electrons to holes through a gate voltage (Novoselov et al. 2004). Also, the quantum Hall effect shows half-integer numbers (Novoselov et al. 2005; Zhang et al. 2005). Because of its fundamental importance in physics as a realization of a relativistic condensed-matter system, as well as its application potentials in next-generation electronics, the research interest in graphene has been rising rapidly (Geim et al. 2007; Castro Neto et al. 2008). (p. 442)

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Fig. 14.1 Unit vectors of graphene in the real space (a) and the reciprocal space (b).

14.2 Electronic structure of graphene

The carbon atoms in graphene are arranged in a honeycomb lattice as shown in Fig. 14.1. The unit cell of graphene contains two geometrically different carbon sublattices A and B. The lattice vectors can be written as a1=a(12,32)a1=a(12,32),, where a is the carbon–carbon lattice constant √a = 1.42 Å . The corresponding reciprocal lattice vectors are b1=4π3a(32,12)b2=4π3a(32,12) and the Brillouin zone also shows a hexagonal shape.

In graphene, the p electrons form the sp2 bonding. The intraplanar interaction between the 2s, 2px, 2py atomic orbitals form the strongly covalent σ orbitals, which give rise to three bonding σ bands (Fig. 14.2) and three anti-bonding σ bands. The out-of-plane pz atomic wavefunction forms weakly van der Waals π bonds, which give rise to the π and π bands that cross the Fermi energy at the six corners of the hexagonal Brillouin zone. (p. 443)

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Fig. 14.2 The band structure of graphene.

The dispersion of the π and π bands is

E(k)=±γ1+4cos(kxa2)cos(3kya2)+4cos2(kxa2),

where γ∼3 eV (Dresselhaus et al. 1981) is the nearest-neighbor hopping integral, a=3a0 with the carbon–carbon lattice constant a0 = 1.42 Å . The valence and conduction bands touch at the six corners of the Brillouin zone. Since each carbon atom contributes one electron, the valence band is completely filled up to the Fermi level, while the conduction band is empty. Because of the zero separation between the bottom of the conduction band and the top of the valence band, graphene is also known as a semi-metal or zero-gap semiconductor.

The most peculiar property of graphene is in the region near the Brillouin zone corners where the valence and conduction bands merge. Expanding the dispersions near the Brillouin zone corner K point, k=K+κ, the dispersion can be written as E(k)=±3a2γ|κ|=±vFκ, where vF=3aγ2. This dispersion relation is analogous to that of relativistic particles E=±m2c4+c2p2 with zero effective mass m = 0, and the speed of light c is replaced by the Fermi velocity ћvF, which is approximately 300 times smaller. Therefore, electrons in graphene are governed by a two-dimensional version of the relativistic theory introduced by Dirac. Because of this similarity, the low-energy electrons in graphene are also described as massless “Dirac fermions” and the points where the valence and conduction bands merge are termed “Dirac points”.

Figure 14.3 shows a schematic drawing of the low-energy dispersions near EF. The peculiar linear dispersion near EF results in many intriguing properties. First, the “Fermi surface” in graphene contains only six points, different from a real surface common in metals. Because of the finite number of points at the Fermi surface, the density of states vanishes at EF. Moreover, electrons in graphene travel with a constant velocity, in contrast to electrons in a semiconductor, where the dispersion shows a quadratic behavior near the top and bottom of the bands. An electron in the latter case is modelled as a quasi-particle with a finite (“massive”) effective mass m*=2(d2Edk2)1 and can be described by non-relativistic theory formulated in Schrödinger’s equation. The effective mass m is usually different from the non-interacting electron mass and this mass renormalization is used to take into account the effects of electron–electron and electron–phonon interactions.

The electronic structure of epitaxial graphene—A view from angle-resolved photoemission spectroscopyClick to view larger

Fig. 14.3 Schematic drawing of the low-energy dispersions near EF.

(p. 444) The completely linear dispersion near EF and the massless Dirac fermions are expected in single-layer graphene only in the case of a perfect crystal. In the presence of additional interactions, e.g. breaking of the carbon-sublattice symmetry, the valence and conduction bands will hybridize; causing a finite gap between the valence and conduction bands. One extreme case is BN, which has a similar hexagonal structure and two completely different sublattices, which results in a gap of up to a few eV (Blase et al. 1995). Similar gap opening can be expected if the potentials on the two carbon sublattices are inequivalent. When a gap exists between the valence and conduction bands, the Dirac fermions will acquire a finite mass. However, despite a small deviation near EF from the linear dispersion, the overall dispersion will still show a linear behavior when moving far enough away from the Dirac point.

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Fig. 14.4 (a–c) Dispersions near the K point for single layer, bilayer and trilayer graphene.

From Partoens and Peeters (2007), copyright American Physical Society.

In the case of bilayer and trilayer graphene, the number of π bands will increase with the number of graphene layers. The overall linear dispersion is still preserved, along with some additional parabolic perturbations near EF (Fig. 14.4; Partoens and Peeters 2007). When the number of graphene layers exceeds 10, the electronic structure of multilayer graphene is basically that of graphite.

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Fig. 14.5 LEED patterns for different stages of the graphene growth at 180 eV: (a) 1 × 1 of SiC, (b) 3 × 3, (c) (3×3)R30 (d) (63×63)R30 (e) sharper 63×63R30. Panel (f) is taken at the same stage as panel (e) but with a lower energy of 130 eV, where the graphene spots (pointed to by the arrows) can be observed.

(p. 445) 14.3 Sample growth and characterization

Graphene research has focused mainly on two types of samples, exfoliated graphene and epitaxial graphene. The exfoliated graphene sample is produced by mechanical exfoliation of graphite flakes followed by deposition on a Si wafer coated with 300 nm of SiO2 (Novoselov et al. 2004, 2005). Exfoliated graphene samples have been widely used in transport measurements as they have very high mobility and its structure makes it easy to apply a gate voltage to tune the charge carriers. However, exfoliated graphene samples have low yield and they are typically small (of the order of 10 μm, much smaller than the size of the synchrotron beam ∼100 μm used in ARPES) and therefore it is difficult to study the electronic structure of exfoliated graphene using ARPES. So far, the electronic structure of graphene has been mostly reported only in the other type of graphene sample, epitaxial graphene. It is well known that thick graphite samples can be grown by cracking of hydrocarbon gas on various metals (McConville et al. 1986; Land et al. 1992) or by thermal decomposition of carbide wafers (Van Bommel et al. 1975; Nagashima et al. 1993). However, it was not until recent years that the single-layer epitaxial graphene sample on SiC was fully characterized (Berger et al. 2004). The advantage of epitaxial graphene is that the samples can be made much larger (mm scale) and therefore they are suitable for ARPES study to extract the electronic structure directly. Technologically, epitaxial graphene is important since the existing Si technology can be incorporated to mass produce epitaxial graphene samples (Berger et al. 2006; de Heer et al. 2007; Kedzierski et al. 2008).

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Fig. 14.6 Structure of epitaxial graphene. The buffer layer grows on top of SiC before a graphene layer is formed.

Graphene samples on SiC wafer can be grown by thermal decomposition of the SiC wafer. The choice of SiC substrate has the advantage that SiC has a bandgap of ∼2.6 eV and therefore its electronic structure does not interfere with that of graphene near EF. There are two terminations for the SiC wafers, Si-terminated and C-terminated. The graphene on the C-terminated 4H-SiC shows more inplane orientational disorder (Hass et al. 2006) than graphene on Si-terminated 6H-SiC. Therefore, graphene on Siterminated face is more favorable for ARPES studies. The 6H-SiC is doped with nitrogen to have a resistivity of ∼0.2Ω cm−1 unless mentioned otherwise. The growth process is monitored with low-energy electron diffraction (LEED) or low-energy electron microscopy (LEEM). The wafer was cut into pieces of a few millimeters in size by using a diamond saw, and cleaned with acetone and isopropyl alcohol in an ultrasonic bath. The clean wafers were then mounted to a Ta sample holder and loaded into an ultrahigh-vacuum preparation chamber with base pressure 1 × 10−10 torr. The wafer was then annealed at 850°C under silicon flux with deposition rate about 3 Å per minute to remove the native surface oxides by the formation of volatile SiO. Figure 14.5 shows the LEED pattern at different stages during the growth process. After the initial cleaning under Si flux, the LEED pattern shows a 3 × 3 reconstruction with respect to the SiC substrate, as has been well documented in the literature (Kaplan 1989; Bermudez et al. 1995; Forbeaux et al. 1998; Starke 1998). A subsequent annealing at 1000°C in the absence of Si flux shows a 1 × 1 pattern. More annealing at 1100°C shows the (3×3)R30 reconstruction. (p. 446)

This corresponds to a 1/3 layer of Si ad-atoms in threefold symmetric sites on top of the outermost SiC layer (Northrup 1995; Owman 1995). Further annealing at 1250°C results in the (63×63)R30 pattern, which indicates the formation of the carbon-rich layer. This carbon-rich layer is formed by carbon atoms arranged in the same structure as graphene. However, this layer forms only the σ bands but no π bands characteristic of single-layer graphene (Emtsev et al. 2007). Therefore, this carbon-rich layer is called a buffer layer (Emtsev et al. 2007; Mattausch et al. 2007; Varchon et al. 2007). Finally, annealing at 1400°C forms the graphene layer on top of the buffer layer (Fig. 14.6).

Even though LEED is a good indicator for the different stages of the growth, the distinction between graphene and the carbon-rich (buffer layer) is not obvious, and LEED is not sensitive to the graphene thickness. Therefore, the sample thickness needs to be characterized using other methods.

Low-energy electron microscopy is a powerful technique to study the surface topography and the dynamics of the growth process (Bauer 1994; Phaneuf et al. 2003), as well as characterizing the sample thickness. LEEM is a surface-sensitive imaging tool that collects the backscattered electrons ranging from 1 to 100 eV. The low energy of the electrons used distinguishes LEEM from other electron microscopies, e.g. transmission electron microscopy (TEM) where the electron energy is typically 100 000 eV. The use of low-energy electrons has the advantage that a large fraction of electrons will be backscattered elastically and that the backscattered electrons in this low-energy range are very sensitive to the physical and chemical properties of the surface. Thus, depending on the energy of the incident electrons, subtle differences in the local atomic structure or composition can result in dramatic reflectivity contrast in LEEM. Another advantage of LEEM over other microscopy is that the images can be taken instantly, allowing real-time study of the dynamics. For example, Fig. 14.7 shows the LEEM image before cleaning the SiC wafer at 850°C under Si flux. The LEEM image shows that after the initial cleaning, the sample surface becomes smoother. The direct visualization of the growth process allows a fine tuning of the growth parameters to achieve high-quality samples, and to obtain graphene samples with various characteristics, e.g. different graphene terrace size.

The electronic structure of epitaxial graphene—A view from angle-resolved photoemission spectroscopyClick to view larger

Fig. 14.7 LEEM images taken before and after annealing the SiC substrate under Si flux with a field of view of 5 μm.

(p. 447)

The electronic structure of epitaxial graphene—A view from angle-resolved photoemission spectroscopyClick to view larger

Fig. 14.8 LEEM images taken at electron energy of 6.6 eV with a 3 μm field of view and the energy scans for the buffer layer, single layer and bilayer graphene.

In addition to studying the surface topography, LEEM can also yield direct information about the graphene sample thickness by studying the quantum-well states on a substrate (Hibino et al. 2007; Ohta et al. 2008). The coherent reflected electrons from the surface and the interface between the graphene and the substrate will interfere with each other and form interference patterns in the reflectivity as a function of electron energy. The number of quantum oscillations directly reflects the number of states, or the thickness of the graphene sample, and can therefore determine the sample thickness. Figure 14.8 shows a LEEM image with intensity ranging from white, gray and black. The energy scan shows zero, one and two minima in the energy scan, allowing us to determine these regions to be buffer layer, single layer and bilayer graphene, respectively. Therefore, the LEEM image shows that the epitaxial graphene has regions of single and bilayer graphene, as well as the buffer layer. The single-layer graphene is the majority and the terrace size of the single-layer graphene can be controlled by tuning the annealing temperature and annealing time.

Additional ways to characterize the sample thickness are: Auger electron spectroscopy (Berger et al. 2004), X-ray photoelectron spectroscopy (Rolling et al. 2006) and angle-resolved photoemission spectroscopy (Ohta et al. 2006).

The electronic structure of epitaxial graphene—A view from angle-resolved photoemission spectroscopy

Fig. 14.9 Schematic drawing of ARPES measurements. A monochromatic light shines on the sample and the photoelectrons are collected by an electron analyzer.

14.4 Electronic structure of epitaxial graphene

Angle-resolved photoelectron spectroscopy (ARPES) is a direct and powerful tool to probe the electronic structure of a material (Damascelli et al. 2003). The basic principle behind this technique is the photoelectric effect. When a beam of light with photon energy hν shines on a clean sample surface, photoelectrons will be emitted from the sample, which are collected by an electron analyzer as a function of kinetic energy Ek and angle θ (Fig. 14.9). The energy is conserved in the photoemission process, therefore from the kinetic energy Ek of photoelectrons and the work function φ, the binding energy of electrons inside the solid can be determined as EB = hvφEk. For single-crystal samples, because of the translational symmetry in the crystal plane and the negligible wave vector of the photons, the inplane momentum p// is preserved. Therefore, the electron binding energy and inplane momentum can be determined as EB=hvϕEkp//=k//=2mEksinθ. Therefore, ARPES can directly measure the dispersion E(k) and map out the Fermi surface. Moreover, ARPES directly measures the single-particle spectral function A(k, ω), which not only contains the band dispersion, but also the self-energy due to different interactions. (p. 448)

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Fig. 14.10 ARPES data taken along the K direction at a photon energy of (a) 80 eV and (b) 100 eV.

Figure 14.10 shows ARPES data taken in single-layer epitaxial graphene along the high-symmetry direction K in a large energy range. The π bands at low binding energy and the σ bands at high binding energy can be clearly observed, similar to the calculated dispersion shown in Fig. 14.2. In addition to the π and σ bands, the bands from the SiC substrate (pointed to by the arrow in panel (b)) can also be observed and the intensity of these bands increases with higher photon energy. The advantage of using semiconducting SiC as substrate is that the SiC bands are at much higher binding energy, and therefore the low-energy π band near EF, which is the main focus of this study, can be easily separated from the substrate.

Figure 14.11 shows the intensity map measured at EF, −0.4 and −1.2 eV taken on single-layer epitaxial graphene near the Brillouin zone corner K. At the EF (panel (a)), the intensity map shows a small circular pocket instead of a single point as expected, suggesting that the sample is doped. This pocket decreases in size when going down in energy and it appears to be a point near −0.4 eV. Beyond −0.4 eV, the size of the pocket increases again and it shows an overall circular behavior. The modulation of the intensity inside and outside the first Brillouin zone is known to be caused by the dipole matrix element in ARPES (Shirley et al. 1995). These constant energy-intensity maps are in overall agreement with the conical dispersion. However, the sample is electron doped and the Dirac point energy ED lies below the Fermi energy EF.

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Fig. 14.11 (a–c) Intensity maps taken at EF, −0.4 eV and −1.2 eV. (d) Schematic drawing of the conical dispersion and the relative positions for data shown in panels a–c.

(p. 449)

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Fig. 14.12 (a) Intensity map as a function of energy and momentum taken along a cut through the K point (black line in the inset). (b) MDCs at energies from EF to −1.5 eV.

The conical dispersion and the electron doping can also be studied from the dispersion. Figure 14.12 shows the dispersion along a line through the K point. Two cones dispersing in opposite directions, one upward and another downward, can be clearly observed. The center point between these two cones, which is defined as the Dirac point energy ED, lies at −0.4 eV. Figure 14.12(b) shows the momentum distribution curves (MDCs), intensity as a function of momentum, for energies between EF and −1.5 eV. The dispersion can be extracted by following the peak positions in the MDCs, which shows an almost linear behavior for both the upper cone and lower cone. Overall, the data presented so far on single-layer epitaxial graphene show that the dispersion is in agreement with the conical dispersion and that the sample is electron doped.

Figure 14.13 shows the evolution of the dispersion with graphene sample thickness. From single-layer to bilayer graphene, the π band splits into two bands as a result of interlayer interaction. In addition, the bilayer graphene sample is also electron doped and there is a finite electron pocket at EF. For trilayer graphene, ED moves even closer to EF, and there is still a finite electron pocket at EF (Ohta et al. 2007; Zhou et al. 2007). Eventually, when the sample thickness is infinite, ED lies almost at EF (Zhou et al. 2006).

The thickness dependence in Fig. 14.13 suggests that the electron doping comes from the interface. To test if the doping is related to the doping of the SiC substrate or the Si atoms at the interface, we measured two bilayer graphene samples on a different substrate 4H-SiC with resistivity of 105Ωcm−1 compared to 0.2Ωcm−1 in the 6H-SiC substrate studied previously. (p. 450)

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Fig. 14.13 Intensity map measured along a line through the K point for (a) single layer, (b) bilayer (c) trilayer epitaxial graphene, and (d) graphite. The arrows mark the position of the Dirac point energy, which moves closer to EF with increasing sample thickness.

Figure 14.14 shows that even though the resistivity of the substrate is different by a factor of 105, ED is still at the same energy of −0.29 eV within an error bar of 20 meV. Figure 14.15 summarized the data taken on graphene samples on both 6H-SiC and 4H-SiC. It is clear tat even though ED changes with graphene sample thickness, it is independent of the doping of the SiC substrate. This comparison shows that the doping of the graphene sample is related to the charge transfer at the interface from the Si atoms (Zhou et al. 2007), and it not determined by the doping of the substrate. This is in agreement with the thickness dependence, since for the thicker graphene sample, it is farther away from the interface, and therefore the change in ED is smaller and ED is closer to EF.

Data presented so far show that epitaxial graphene samples are electron doped. The electron doping is most likely associated with the charge transfer from the Si atoms at the interface and not related to the doping of the substrate. The dispersions show an overall conical behavior. However, near the Dirac point energy (Fig. 14.13(a)), there is an anomalous vertical region. To understand this vertical region near ED, a more detailed analysis of the dispersion near ED is required.

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Fig. 14.14 Data taken through the K point for bilayer graphene (a) on more conductive 6H-SiC substrate with resistivity of 0.2Ωcm−1 (b) on more insulating 4H-SiC substrate with resistivity of 105Ωcm−1.

(p. 451) 14.5 Gap opening in single-layer epitaxial graphene

Figure 14.16 shows a more detailed analysis of the data near ED at the Brillouin zone corner K. The dispersions can be extracted from the energy-distribution curves (EDCs), intensity scans as a function of energy for fixed momentum. In the EDCs, there are two peaks that correspond to the valence and conduction bands. It is interesting to note that even at the K point, where these two bands are expected to meet for a perfect conical dispersion, there are still two peaks in the EDCs. This presence of two peaks at the K point suggests that the valence and conduction bands do not merge at the K point. Instead, there is a finite separation between these two bands. To extract the dispersion for both the valence and conduction bands, we fit EDCs with two Lorentzians multiplied by the Fermi–Dirac function. The extracted dispersions are shown as white lines in panel (a), which clearly show a finite separation of ∼0.26 eV between the valence and conduction bands. This anomalous region between the valence and conduction bands can also be observed in the MDCs shown in panel (c). Away from ED, two peaks are clearly observed in the MDCs, while in an extended region between −0.3 eV and −0.5 eV, only one peak is observed in the MDC and the peak position does not change with energy. The non-dispersive MDC peak is a typical ARPES feature of the gap (Shen et al. 2005). Panel (d) shows the angle-integrated intensity curve, which is proportional to the density of states. At high binding energy, the angle-integrated intensity curve shows a linear behavior, while near the Dirac point energy, there is a dip suggesting suppression of the density of states below the expected linear behavior. This is in agreement with the gap opening near ED. In summary, Fig. 14.16 shows that the dispersion near ED deviates from the linear behavior and there is a finite gap opening between the valence and conduction bands. One interesting observation is that there is still finite intensity inside the gap region.

The electronic structure of epitaxial graphene—A view from angle-resolved photoemission spectroscopy

Fig. 14.15 Summary of the Dirac point energyED as a function of graphene sample thickness.The open symbols are taken on graphene on 6H-SiC and the filled symbol is taken on a bilayer graphene sample on 4H-SiC.

Figure 14.17 shows how the dispersions evolve across the K point. Panels (a,b) show the intensity maps at EF (above ED) and −1 eV (below ED), respectively. Above ED, the intensity is stronger outside the first Brillouin zone, while below ED the intensity is stronger inside the first Brillouin zone. This intensity modulation is related to the dipole matrix element in graphene (Shirley 1995) and can be utilized to identify the exact K point. Panels (c–h) show the data taken before and after crossing the K point, with a very fine step of 0.05 deg close to the K point, which corresponds to a step of 0.0026 Å−1 to make sure to catch the K point as accurately as possible. Before crossing the K point (panel (c)), the data shows a stronger intensity for the valence band, while after crossing the K point (panels (g,h)), the data show a stronger intensity for the conduction bands, in agreement with the intensity modulation in panels (a,b). In all the panels, the valence and conduction bands can be followed from the EDC peak positions. The most important information comes from the EDC at ky = 0, where the separation between these two bands is the smallest in each panel. Panel (i) shows the EDCs at ky = 0. Two peaks can be identified in the EDCs, and the intensity modulation between these two peaks switches before and after crossing the K point. This intensity modulation enables us to identify that the K point is for cut (e), since for this cut the EDC shows an almost identical intensity for the two peaks and the gap is smallest. The gaps for the various cuts taken can be plotted a function of the momentum kx, as shown in panel (j). The consistent trend of the gap with the minimum value of ∼0.26 eV at the K point shows that the gap is not an artifact due to sample misalignment, therefore the upper cone and lower cones are separated by a finite amount and there is a bandgap of ∼0.26 eV in single-layer graphene. In addition, a finite intensity is always observed inside the gap region for all the cuts.

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Fig. 14.16 (a) Data taken through the K point. The white lines show the dispersions extracted from panel (b). (b) EDCs taken from k0 to k12 as labelled at the bottom of panel (a). The dots are the raw data and the black curves are the fits using two Lorentzians multiplied by the Fermi–Dirac function. (c) MDCs from EF to −0.8 eV. (d) Angle-integrated intensity as a function of energy for data shown in panel (a).

(p. 452)

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Fig. 14.17 (a, b) Intensity maps taken at EF (a) and −1 eV (b) near the K point. The vertical lines show the position of the cuts for data shown in panels (c–h). (c–h) Data taken near (c,d,f,g,h) and through (panel e) the K point. The dotted lines are guides for the eye for the dispersions of the valence and conduction bands. (i) EDCs taken at ky = 0. The tick marks above the EDCs label the EDC peak positions. (j) Plot of the gap as a function of kx when approaching the K point.

(p. 453)

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Fig. 14.18 (a) Schematic drawing of the cuts for data shown in panels (b,c). The cartoon in the inset shows schematically the gap opening for the cut away from the K point even when there is no gap at the K point. (b,c) Data taken through and off the k point. The dotted lines are the extracted dispersions from the MDCs. The arrows mark the position of the top valence band and the bottom of the conduction band. (d) Comparison of the dispersions for the cut through and off the K point. (e) Extracted MDC width as a function of energy for data shown in panels (b,c).

The dispersions near the K point can also be studied by measuring along another geometry (see Fig. 14.18(a)), which shows asymmetric intensity for the dispersions on the two sides of the K point. This geometry has the advantage of showing only one branch of the dispersion, thereby enabling an easier fit of the MDCs to extract the dispersion and the MDC width, with the additional disadvantage that the Dirac point cannot be envisioned directly. In addition, one caveat of the MDC analysis is that the presence of a peak does not necessarily mean the presence of a quasi-particle, therefore one also needs to be careful about the interpretation of the MDC dispersions (Zhou et al. 2008b). Panels (b) and (c) show two cuts, one through the K point and another one off the K point. In panel (b), the top of the valence band and the bottom of the conduction band for the cut through the K point (panel (b)) are determined from the symmetric geometry discussed previously and are labelled by the two gray arrows in panel (b). The black arrows in panel (c) label the top and bottom of the bands determined for the cut off the K point. The data show an overall similarity between these two cuts with a suppression of intensity in the region near the Dirac point energy. The extracted MDC dispersions show an even larger deviation from the linear dispersion for the cut off the K point. In addition, the MDC width shows an anomalous region near ED, where multiple peak structures appear. This anomalous region has been discussed previously by Bostwick et al. (2006) and was attributed to electron–plasmon interaction. However, it is interesting to note that energies for the top of the valence band and bottom of the conduction band coincide with the two peak positions in the anomalous region for both cuts (b) and (c), and the amplitude of the peaks is larger for the cut off the K point. This suggests that the anomalous region near ED has a similar origin for the cuts through and off the K point (Zhou et al. 2008b). In the latter case, it is known from the conical dispersion that a gap is definitely expected. This is in agreement with previous EDC analysis of Fig. 14.16 and Fig. 14.17, where a gap is present.

Figure 14.19 shows that the anomalous peaks in the MDC width near ED can be explained by the gap (Zhou et al. 2008b). Panels (a) and (b) show the simulated data with the dipole matrix element that enhances one branch of the dispersion, similar to the case in graphene. Applying a similar MDC fit and extracting the MDC width, the MDC width shows two peaks in the gap region and the peak is larger for the larger gap. This similarity with respect to the experimental data discussed in Fig. 14.18 shows that the gap opening is the most likely origin for explaining the deviation near ED and the anomalous peaks in the MDC width, and not due to electron–plasmon interaction. (p. 454)

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Fig. 14.19 (a,b) Simulation of the conical dispersions with a gap of 150 meV and 400 meV with added dipole matrix element to reproduce similar intensity pattern as the experimental data on epitaxial graphene. (c) Extracted MDC width from the dispersions.

14.6 Possible mechanisms for the gap opening

There are a few possible mechanisms that can open up a gap in graphene: quantum confinement (Nakada et al. 1996; Brey et al. 2006; Son et al. 2006; Han et al. 2007; Nils et al. 2007; Nilsson et al. 2007), mixing of the states between the K and Kʹ points induced by scattering, and hybridization of the valence and conduction bands caused by breaking of carbon sublattice symmetry (Manes et al. 2007). We will examine all of these possibilities and we think that the most likely scenario is the breaking of the carbon-sublattice symmetry.

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Fig. 14.20 Schematic diagram showing the electronic states for carbon nanotubes.

(p. 455) 14.6.1 Quantum confinement

The gap induced by quantum confinement is well known in the case of carbon nanotubes (Saito et al. 1992) and has been recently extended to graphene nonoribbons (Nakada et al. 1996; Son et al. 2006; Han et al. 2007; Trauzettel et al. 2007). Carbon nanotube is a sheet of rolled graphene, therefore additional boundary conditions apply to carbon nanotube. When the graphene sheet is rolled along a certain direction to form carbon nanotubes, the electronic states will be quantized along this direction (Fig. 14.20). For certain directions where the bands do not cross the Dirac point, the carbon nanotube will be a semiconductor with a finite size gap (Saito et al. 1992). The electronic states near EF are determined by the intersection of the allowed K points with the dispersion cones at the K point. The size of the gap increases as the carbon nanotube becomes smaller, and scales inversely with the width of the ribbons. The dependence of the gap size on the graphene ribbon width has been recently shown by Han et al. in exfoliated graphene (Han et al. 2007). It is important to note that for ribbons as small as 10 nm, the gap size can be as big as 200 meV, while for ribbons larger than 30 nm, the gap would be smaller than 10 meV. Therefore, it is important to measure the gap in our graphene samples with various graphene ribbon sizes to test whether the scenario of quantum confinement can account for the large gap observed here.

The electronic structure of epitaxial graphene—A view from angle-resolved photoemission spectroscopy

Fig. 14.21 LEEM image with a 2-μm field of view taken on a graphene sample to show how the graphene terrace size is quantified. The gray, black and white regions are the single-layer, bilayer graphene and the buffer layer. The tick marks along the diagonal direction label the edges of the single-layer graphene (gray area), which were used to quantify the single-layer graphene terrace size.

Single-layer graphene samples with various terrace size can be obtained by controlling annealing temperature and annealing time. The real-time observation of the growth process is a unique advantage to control the ideal growth conditions and obtain graphene samples with various sizes. As discussed previously in Section 14.2, LEEM is a powerful tool to study the surface topography and identify the single-layer graphene regions. Therefore, the graphene terrace size can be directly measured from the LEEM image. Figure 14.21 shows a LEEM image with a majority of single-layer graphene with smaller percentage of bilayer graphene and buffer layer. The graphene terrace size and the error bar is quantified by taking the average and the standard deviation of the graphene terrace sizes crossed by the two lines along the diagonal direction.

Figure 14.22 shows the LEEM images for a few graphene samples with various single-layer graphene terrace sizes. The samples contain a majority of single-layer graphene with different terrace sizes, along with a much smaller fraction of bilayer graphene and buffer layer. The single-layer graphene regions are the focus here. The single-layer graphene terrace size in each panel is characterized using the method described above. In these samples, with increasing annealing, the graphene terrace size can be controlled from 50 nm (panel (e)) to 180 nm (panel (a)). After the LEEM measurements, the same samples were transported to the ARPES chamber. After annealing at 700°C to remove the gas adsorbed on the graphene surface, the samples were measured with ARPES to yield direct information about the electronic structure. Figure 14.23 shows the corresponding ARPES data for the same samples measured in the same geometry. Following the same procedure, the gap can be determined by measuring the distance between the valence band and conduction band extracted from the EDC peak positions. When the graphene terrace size decreases from panel a to panel e, the gap increases slightly. (p. 456)

The electronic structure of epitaxial graphene—A view from angle-resolved photoemission spectroscopyClick to view larger

Fig. 14.22 LEEM image for various graphene samples width different graphene terrace sizes of (a) 57 nm, (b) 85 nm, (c) 95 nm, (d) 121 nm, (e) 179 nm. The circle in each panel marks the characteristic width of the graphene terrace.

Figure 14.24 shows the direct correlation of the gap size measured from ARPES with the graphene terrace size obtained from the LEEM measurements (Zhou et al. 2008a). It is clear that the gap slightly increases for the smallest graphene terrace size. However, the gap does not change within the experimental uncertainty even when the graphene terrace size exceeds 150 nm. This is in contrast to the gap induced by quantum confinement in the exfoliated graphene nanoribbons where the gap is below 10 meV when the graphene terrace size is larger than 30 nm. This direct comparison shows that even though quantum confinement can contribute to enhance the gap size, quantum confinement cannot explain the large gap of 180 meV still observed in largest graphene terrace size. Therefore, quantum confinement is not the main mechanism for the gap opening in epitaxial graphene (Zhou et al. 2008a).

14.6.2 Inter-Dirac-point scattering

Another possible way to open up a gap is to hybridize the electronic states at K and Kʹ (Manes et al. 2007) as schematically shown in Fig. 14.25. This requires breaking of the translational symmetry. It is known that in graphene there are reconstructions of 6 × 6 and (6✓3 × 6✓3)R30° (Tsai et al. 1992; Forbeaux et al. 1998). However, the scattering vectors related with these two constructions are much smaller than the K–Kʹ distance, which is required to mix the states at K and Kʹ. Higher-order scattering process involving consecutive small scattering vectors is weak in general and is an unlikely source for the gap opening. Impurity scattering can also mix the states at K and Kʹ (Rutter et al. 2007). However, this would give rise to a gap that strongly depends on impurity concentration. This is in contrast to our findings, where the gap is similar for all the samples studied.

The electronic structure of epitaxial graphene—A view from angle-resolved photoemission spectroscopyClick to view larger

Fig. 14.23 Corresponding ARPES data taken along a line through the K point for samples shown in Fig. 14.22 with various graphene terrace sizes.

From panel (a) to panel (e), the single-layer graphene terrace size decreases. The white lines show the dispersions extracted by fitting the EDCs.

(p. 457)

The electronic structure of epitaxial graphene—A view from angle-resolved photoemission spectroscopy

Fig. 14.24 Gap size measured from ARPES as a function of single-layer graphene terrace size measured from LEEM. The dotted line is the gap measured in exfoliated graphene reported by Han et al. (2007), copyright American Physical Society, where the gap is induced by quantum confinement.

14.6.3 Breaking of the carbon-sublattice symmetry

The most likely scenario to explain the gap that we observed in epitaxial graphene is the breaking of the carbon sublattices (see Fig. 14.26). We note that a perfectly conical dispersion is expected only in the case of perfect single-layer epitaxial graphene, where the potentials on the two carbon sublattices are the same. When this symmetry breaks down, the hybridization between the valence and conduction bands can open up a gap. This has been well known in the extreme case of BN, which has a similar honeycomb structure and completely different sublattices in the unit cell. In the case of BN, the bandgap can be as big as 5.8 eV (Blase et al. 1995).

In the case of single-layer epitaxial graphene, since the sample is grown on top of the SiC substrate and there is a buffer layer between graphene and the SiC substrate, the buffer layer can break the symmetry between the two carbon sublattices. A prediction of this scenario is that breaking of the sixfold rotational symmetry of graphene near the Dirac point energy. Figure 14.27(e) shows the calculated intensity map at ED using a tight-binding model in the

The electronic structure of epitaxial graphene—A view from angle-resolved photoemission spectroscopy

Fig. 14.25 Schematic drawing of inter-Dirac-point scattering between the K and Kʹ points to induce a gap at the Dirac point.

(p. 458)

The electronic structure of epitaxial graphene—A view from angle-resolved photoemission spectroscopyClick to view larger

Fig. 14.26 Schematic drawing to show that when the two carbon sublattices have different potentials, the valence and conduction bands can hybridize to open a gap at the Dirac point.

extreme case when only one of the two carbon sublattices have non-zero potential. The potential modulation imposed by the (6✓3 × 6✓3)R30° reconstruction has been added as a perturbation to the Hamiltonian (Zhou et al. 2007). For energy well above or below ED, the symmetry is restored. Figures 14.27(a–d) show the constant energy-intensity maps measured at EF, ED and below ED. The dominant feature in all these intensity maps is the small pockets centered at the K points. In addition, there are six weaker replicas surrounding each K point. The intensity of these replicas is only ∼4% that of the main peak at the K point. These replicas are associated with the (6✓3 × 6✓3)R30° observed in LEED. At EF (panel (a)), the six replicas show similar intensity, while near ED (panel (b)), three of the six replicas (pointed to by the arrows) are enhanced, suggesting that the sixfold symmetry is broken near ED. At higher binding energy far away from the Dirac energy (panel (d)), the six replicas show similar intensity again, showing the restoration of the sixfold symmetry. Our experimental data shown in panels (a–d) are in good agreement with theoretical predictions that show the breaking of the sixfold symmetry near ED. We note that scanning tunnelling microscopy STM measurements (Brar et al. 2007) on epitaxial single-layer graphene did not report the evidence

The electronic structure of epitaxial graphene—A view from angle-resolved photoemission spectroscopyClick to view larger

Fig. 14.27 Constant energy intensity maps taken at (a) EF, (b) −0.4 eV, (c) −0.8 eV, (d) −1.0 eV. (e) Calculated ARPES intensity map at ED in the presence of symmetry breaking on the two carbon sublattices. The arrows in panel (b) point to the three replicas that are enhanced compared to the other three.

Note that the intensity for the main spots at the K points is saturated to enhance the weaker replicas surrounding the K points.

(p. 459)

The electronic structure of epitaxial graphene—A view from angle-resolved photoemission spectroscopyClick to view larger

Fig. 14.28 (a) Intensity map at ED. The dotted vertical lines mark the positions where the MDCs in panel (b) are taken and the markers (∗) label the position of the EDCs show in panel (c). (b) MDCs taken from kx1 to kx8. (c) EDCs taken from kK, kΓ, ka to ki.

Note that the EDCs at K and are scaled by 0.03 and 2, respectively.

of sixfold symmetry breaking. However, the STM data are taken far away from the Dirac point, where it is known that the effect of the symmetry breaking is much weaker and difficult to observe. Therefore, the lack of evidence of sixfold symmetry breaking in STM studies does not contradict our ARPES results.

In addition, there are some interesting structures in the intensity maps near ED. In particular, we observed hexagonal patterns where the intensity is enhanced (see Fig. 14.27(b)). There are two observations associated with these additional patterns. First, the center mid-sized hexagon around Γ (gray dotted line in panel (b)) almost overlaps with the first Brillouin zone of SiC. Second, all other hexagons (gray broken lines in panel (b)) are not regular, i.e. the six sides that form the hexagon do not have the same length. However, they all pass through the K points. These two observations suggest that the states associated with the hexagonal pattern are related to the buffer layer between the SiC substrate and the graphene.

Figure 14.28 shows a more careful characterization of the in-gap states. Figure 14.28(b) shows the MDCs at ED for momenta from kx1 to kx8. Three non-dispersive peaks are observed in the MDCs, which correspond to the three horizontal segments of the additional hexagonal patterns. This shows that the mid-gap states have maxima along the hexagonal patterns. Panel (c) shows the EDCs taken at various points in the Brillouin zone. At the K point, the EDC shows two peaks separated by a finite amount, which correspond to the top of the valence band and the bottom of the conduction band, in agreement with the gap. Along the hexagonal pattern, the EDCs show a very broad peak from EF to −0.8 eV, while away from the hexagonal pattern (see, e.g. EDC at the point), the broad peak in the EDC can still be observed. However, the intensity is only half of that along the additional hexagons. The analysis of Fig. 14.28 shows that there are additional states that are peaked at the Dirac-point energy, and these mid-gap states form interesting hexagonal pattern in the momentum space. The exact origin of these mid-gap states still requires further studies.

The electronic structure of epitaxial graphene—A view from angle-resolved photoemission spectroscopyClick to view larger

Fig. 14.29 (a) Dispersion near the K point from ab-initio calculation. (b) Contour of the potential on the graphene layer generated by the buffer layer.

From Kim et al. (2008).

(p. 460) The proposed mechanism of breaking carbon-sublattice symmetry to explain the gap opening in epitaxial graphene is also supported by recent ab-initio calculation (Kim et al. 2008), which shows a gap near ED with a gap value similar to ARPES measurements. The calculation shows that the graphene layer follows the atomic structure of the buffer layer underneath and there is a corrugation of the height of the graphene layer, which results in a 140-meV average potential difference in the two carbon sublattices. Moreover, mid-gap states are also predicted, which are caused by the interlayer coupling between the π states in graphene and the localized π states in the buffer layer. This is in agreement with the observation of anomalous intensity in the gap region. However, further studies are still needed to understand why the mid-gap states form such hexagonal patterns in the momentum space as discussed above.

Our proposed mechanism of graphene–substrate interaction as the origin for the opening of the bandgap has important implications, since engineering the bandgap in graphene is an important topic for its applications. It was predicted that when graphene is grown on BN substrate, a gap can also be expected and the gap depends on the orientation of graphene relative to the BN substrate as well as the distance between graphene and the substrate (Giovannetti et al. 2007). If one can tailor graphene–substrate interaction by growing graphene on different substrates, it is possible to engineer the bandgap of graphene in a wider range. Further study to move the Fermi energy inside the gap region with hole dopants to eventually make graphene a semiconductor is also an important topic.

14.7 Conclusions

In summary, we have studied the electronic structure of epitaxial graphene. Clear deviations from the conical dispersions are observed near the Dirac-point energy, and interpreted as a gap opening due to graphene–substrate interaction. This points to graphene–substrate interaction as a promising route to engineer the bandgap in graphene. Even though it might take a long time before graphene’s full application potentials can be fully realized, graphene is definitely a very intriguing system and there is still a lot more to be explored.

(p. 461) Acknowledgments

We thank A.V. Fedorov, D.A. Siegel, G.-H. Gweon, F. El Gabaly, A.K. Schmid, K.F. McCarty and J. Graf for experimental assistance, P.N. First, W.A. De Heer for helping with the graphene samples in the initial stage of this project, D.-H. Lee, F. Guinea, and A.H. Castro Neto for useful discussions.

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